Question

Lennon has 6 hours to spend in Ha Ha Tonka State Park. He plans to drive around the park at an average speed of 20 miles per hour, looking for a good trail to hike. Once he finds a trail he likes, he will spend the remainder of his time hiking it. He hopes to travel more than 60 miles total while in the park. If he hikes at an average speed of \(1.5\) miles per hour, which of the following systems of inequalities can be solved for the number of hours Lennon spends driving, \(d\), and the number of hours he spends hiking, \(h\), while he is at the park? A) \(1.5 h+20 d>60\) \(h+d \leq 6\) B) \(1.5 h+20 d>60\) \(h+d \geq 6\) C) \(1.5 h+20 d<60\) \(h+d \geq 360\) D) \(20 h+1.5 d>6\) \(h+d \leq 60\)

Short answer

The short answer is: Option A: \(1.5 h+20 d>60\) \(h+d \leq 6\)

Step-by-step solution

\(h + d \leq 6\). It represents the sum of hours spent driving and hiking. Now, we need to find an inequality for the distance constraint. #Step 2: Distance constraint# Lennon wants to travel more than 60 miles total in the park. While driving, he covers 20 miles per hour, so when driving the distance covered is \(20d\). While hiking, he covers 1.5 miles per hour, so when hiking the distance covered is \(1.5h\). The total distance is the sum of driving and hiking distances:

\(1.5h + 20d > 60\). It represents that the total distance traveled should be greater than 60 miles. #Step 3: Compare the options with the derived inequalities# Now, we will compare the derived inequalities with the given options, looking for a match. The derived inequalities are: 1. \(1.5h + 20d > 60\) 2. \(h + d \leq 6\) Option A: 1. \(1.5 h+20 d>60\) 2. \(h+d \leq 6\) Option B: 1. \(1.5 h+20 d>60\) 2. \(h+d \geq 6\) Option C: 1. \(1.5 h+20 d < 60\) 2. \(h+d \geq 360\) Option D: 1. \(20 h+1.5 d>6\) 2. \(h+d \leq 60\) Comparing the derived inequalities with the options, we see that Option A matches the derived inequalities. Therefore, the correct choice is:

Answer

Option A: \(1.5 h+20 d>60\) \(h+d \leq 6\)

Key concepts

Systems of Inequalities

When tackling SAT math problems, understanding systems of inequalities is vital.
In this exercise, Lennon has two simultaneous conditions or constraints governing his actions in the park: the time constraint and the distance constraint.

• The time constraint: The total hours spent hiking (\( h \)) and driving (\( d \)) should not exceed 6 hours, hence the inequality: \[ h + d \leq 6 \]
• The distance constraint: Lennon wants to cover more than 60 miles in total. The distance traveled while driving is \( 20d \) (since he drives at 20 miles per hour), and the distance while hiking is \( 1.5h \) (hiking at 1.5 miles per hour). The inequality is represented as: \[ 1.5h + 20d > 60 \]

By using these inequalities, we create a system of inequalities. Solving means finding values of \( d \) and \( h \) that satisfy both conditions simultaneously. This ensures Lennon meets his criteria for both time and distance.

Distance-Speed-Time Relationship

Understanding the distance-speed-time relationship helps solve many real-world problems, including those found on the SAT. This fundamental principle is expressed by the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \]Let's break this formula down in the context of the problem:- **Driving:** Lennon drives around the park at a speed of 20 miles per hour. For each hour he spends driving (\( d \)), the distance he covers is \( 20 \times d \) miles.- **Hiking:** Hiking is more leisurely at a speed of 1.5 miles per hour. For every hour Lennon hikes (\( h \)), he covers \( 1.5 \times h \) miles.By setting these expressions equal to Lennon's total travel distance desire (>60 miles), we directly relate his time allocations to his driving and hiking speeds. Thus, seamlessly connecting his activity duration and the travel achievement.

SAT Problem-Solving Techniques

Solving SAT math word problems, like Lennon's scenario, requires strategic problem-solving techniques. As you practice, consider these steps to enhance your problem-solving skills:1. **Understand the Problem:** Read the question carefully. Identify what is being asked, and underline key information. - For instance, in this problem, note Lennon has 6 hours and desires to travel more than 60 miles.2. **Translate Words to Math:** Convert the words into equations or inequalities. - Here, devise inequalities for both time and distance constraints. 3. **Manipulate Equations:** Re-examine your equations to understand what they reveal about potential solutions. - Ensuring you comprehend how \( h + d \leq 6 \) and \( 1.5h + 20d > 60 \) represent the scenario is key.4. **Compare with Options:** When given multiple choices, compare them against your derived equations. - As seen in this problem, compare derived inequalities with options, finding that 'Option A' is correct.Practicing these methods will build your confidence and efficiency in tackling SAT math word problems. Approach each problem systematically, and you're more likely to succeed.